Multi-objective optimization design of aerodynamic and infrared characteristics of multi-stream serpentine nozzle

A multi-objective optimization design of multi-stream serpentine nozzle was carried out to improve the aerodynamic performance and reduce the infrared radiation signal. A numerical simulation was carried out to investigate the effects of nozzle pressure ratio (NPR), Aspect ratio (AR) and Ratio of length and diameter (L/D). Computational fluid dynamics (CFD) simulation was employed to study the aerodynamic performance of the multi-stream serpentine nozzle, and the narrow-band method with discrete transfer method were used to calculate its infrared radiation intensity distribution characteristics. A 3-factor, 21-level orthogonal table was designed based on the orthogonal experimental method, and multi-stream serpentine nozzle with different geometrical parameter and operating conditions were designed to obtain the aerodynamic performance and infrared radiation intensity data of each experimental sample point. Radial Basis Function (RBF) neural network and Multi-Objective Particle Swarm Optimization algorithm were used to optimize the geometry and operating conditions with better aerodynamic performance and weaker infrared signal. The result shows that the Pareto solutions obtained by the optimization have good accuracy compared with the numerical simulation results, with an aerodynamic performance error of 0.02% and an infrared radiation intensity error of 0.24%.


Introduction
With the rapid development of infrared detection and guidance technology in recent years, increasingly advanced infrared guided weapons restrict the living space of aircraft, and aircraft must face increasingly severe infrared threats.Therefore, it is very important to carry out infrared suppression research and improve the infrared stealth ability of combat aircraft.The exhaust system is the main source of the infrared characteristics of military aircraft in the 3-5μm band, especially the high-temperature components and exhaust gas.When the flight Mach number is less than 1.5, the two contribute more than 90% of the infrared radiation intensity [1].Significantly reducing the infrared and radar signatures of the engine exhaust system can significantly improve the stealth capability of the aircraft [2].
Adaptive cycle engine (ACE) is considered to be an ideal power plant for next-generation aircraft due to its outstanding performance advantages in aircraft thermal management and power extraction [3][4] .ACE essentially belongs to the category of variable cycle engine (VCE).Compared with the VCE of other schemes, ACE has a unique multi-stream structure, and its multiple streams make it more capable of adjusting the bypass ratio than the traditional dual-channel turbofan engine, so that it can be realized on the same engine.A variety of different thermal cycle working modes reconcile the contradiction between high unit thrust and low unit fuel consumption of the engine, so that the engine can automatically adapt and meet the performance requirements of the aircraft mission and the external environment [5].
In order to adapt to the multi-flow structure of the adaptive cycle engine and reduce its infrared radiation intensity, the multi-stream Serpentine nozzle came into being.On the one hand, multi-stream Serpentine nozzle can adjust the flow rate of the third airflow in different ACE modes; on the other hand, the multi-stream Serpentine nozzle configuration can effectively reduce the infrared radiation intensity of the engine, so the multi-stream Serpentine nozzle is a key component to adapt the exhaust system to ACE and realize the ultra-stealth performance of the next generation fighter [6][7][8][9] .The U.S. Air Force launched the Multi-purpose Affordable Advanced Turbine Engine Program (VAATE) in 2007, and its two sub-programs, Adaptive Multi-purpose Engine Technology (ADVENT) and Adaptive Engine Technology Development (AETD) [10], Both adopt the same multi-stream S-curved supersonic nozzle structure, as shown in Figure 1.At present, researchers' research on the multi-stream Serpentine nozzle mainly focuses on the flow and noise characteristics of the multi stream or Serpentine nozzle.Magstadt [11] of Syracuse University in the United States designed a multi-ply Multi-Aperture Rectangular Single Expansive Ramp Nozzle (MARS), and its aeroacoustic characteristics were studied through experiments.Spectrum Energy Company Rusher [12] studied the influence of mainstream heating on the flow structure and far field effect of noise through numerical simulation.Stack [13] of the Ohio State University at Austin used the LES of the turbulent inlet boundary to analyze the flow characteristics of the wake jet, shock wave and shear layer in detail under design conditions, and Hromisin [14] of the Pennsylvania State University investgated the aeroacoustic characteristics of the axisymmetric single-flow convergent-divergent nozzle and the multi-flow supersonic nozzle are studied under the same thrust with the experiment.For the multi-stream supersonic nozzle, the researchers studied its flow field characteristics, aeroacoustic characteristics and unsteady flow characteristics through numerical simulation and experiments [15][16][17][18][19][20][21] .As for serpentine nozzle ,researchers carried out a variety of design methods for the Serpentine nozzle [22][23][24] , and a lot of research on the flow characteristics of Serpentine nozzles, focusing on thrust vector performance and flow characteristics [25][26] , less research on the infrared radiation characteristics of multi-flow Serpentine nozzles.
The structure of the multi-stream Serpentine nozzle is more complicated than that of the general Serpentine nozzle, and there are more configuration parameters.Different structural parameters will affect the aerodynamic performance and infrared radiation characteristics of the multi-stream Serpentine nozzle.In the design process of multi-stream Serpentine nozzle, the design goals of its aerodynamic performance and infrared stealth performance are often contradictory.In order to obtain the best aerodynamic performance, when designing the multi-stream Serpentine nozzle, it is necessary to minimize the change of the flow direction of the airflow in the nozzle, that is, to minimize the lengthto-diameter ratio of the multi-stream Serpentine nozzle and the Aspect ratio to reduce localized losses along the airflow.However, the change of length-to-diameter ratio and aspect ratio will affect the wall temperature distribution and gas mixing intensity of the nozzle, thus affecting the infrared radiation characteristics of the multi-stream Serpentine nozzle.Different design parameters will inevitably change the infrared radiation characteristics while affecting the aerodynamic performance.Aiming at this problem, this paper mainly studies the aerodynamic/infrared multi-objective optimization method of multi-stream Serpentine nozzle.

Nozzle geometry
The multi-stream Serpentine nozzle used in this paper is mainly composed of three parts, the first part is the Serpentine nozzle section, which contains the core and bypass of the engine, the second part is the third bypass section, which is the circulation channel of the third duct airflow, and the third part is the mixing section, in which the airflow of the Serpentine nozzle section and the third bypass section of the mixing section are mixed.For the first part of the S-curve segment, it is a cross-sectional sweep-out surface perpendicular to the centerline and varying in area along the streamline direction, where the cross-section transitions from the axisymmetric circular inlet to the binary rectangular exit, and the centerline consists of two S-shaped curves.The two central lines are constructed based on the Lee relation and the specific equations of the three curves are as follows: Where ΔY is the offset, that is, the projection of the distance from the beginning to the end of the curve in the y-axis direction; L is the total length of the curve along the x-axis; 1), 2), and 3) indicate the change law of "uniform", "front slow and then urgent", and "front urgent and backward" respectively.
The third duct structure of the multi-stream serpentine nozzle studied in this paper is shown in Figure 2(a).From the entrance section of the third duct to the exit section of the second bend, the third ducts on both sides are slightly contracted so that the incoming flow of the third duct can smoothly merge into the main flow; from the exit section of the second bend to the outlet section of the nozzle, the outer walls of the upper and lower sides of the three ducts are slightly flared, so that the straight section becomes an expanded section to meet the needs of mixing and acceleration.
Figure 2(b) shows the schematic diagram of the parameters of the multi-stream Serpentine nozzle and the geometric model.The main parameters changed are the length-diameter ratio(L/D), the nozzle outlet aspect ratio(AR), and nozzle pressure ratio(NPR) when the Aerodynamic /Infrared multiobjective optimization study of the multi-stream Serpentine nozzle is carried out.The range of the design parameters are showed in Table 1.The aerodynamic target for this article is Cfg, which is the thrust coefficient of the nozzle.

CFD simulation method
The wall temperature distribution, gas temperature distribution, pressure distribution and component concentration distribution of the multi-stream Serpentine nozzle will affect the infrared radiation characteristics of the nozzle, so the calculation of infrared radiation characteristics of the multi-stream Serpentine nozzle based on the discrete transfer method is based on the flow field calculation.The temperature distribution of the wall of the multi-stream Serpentine nozzle, the temperature and pressure distribution of the gas, and the molar fraction of CO2, H2O, CO and other gas components were obtained through the flow field calculation, and these parameters were imported into the infrared radiation characteristic calculation program for calculation.In order to improve the calculation efficiency, this paper uses half of the multi-stream Serpentine nozzle surface symmetrical as the flow field to calculate the geometric surface, and the mesh is shown in Figure 4.The near-wall mesh is encrypted.After the verification of grid independence, when the calculated grid amount reaches about 6.2 million, the static pressure distribution of the wall of the multi-stream Serpentine nozzle no longer changes significantly.
In this paper, the commercial CFD software FLUENT is used to solve the three-dimensional Reynolds average Navier-Stokes (N-S) equation, and the SST k-ω turbulence model is used to carry out the numerical calculation of the flow field characteristics of multi-stream Serpentine nozzle.

Figure 3.
The mesh of multi-stream serpentine nozzle.Boundary conditions are important factors affecting the results of the flow field, and improper boundary conditions will lead to errors in the numerical simulation results, and even lead to the divergence of the entire calculation results and fail to obtain effective results.The boundary conditions mainly involved in the calculation model of this study include core and bypass inlet boundary, far-field inlet boundary, far-field outlet boundary, third bypass inlet boundary and solid wall boundary.Among them, the core and bypass boundary is the pressure inlet boundary, the main component is divided into engine gas, and the inlet pressure will change with the change of NPR; The far-field inlet boundary is the pressure inlet boundary, the total pressure P*=1atm, the total temperature T*=300K, the components are N2 and O2 of the standard atmosphere, and the incoming flow velocity V*=0.05Ma;The total pressure of the far-field outlet boundary P*=1atm, the total temperature T*=300K; The total pressure of the third duct inlet P*=1.893atm,T*=300K.

Infrared radiation simulation method
Considering the infrared radiation generated by the wall of the nozzle and the gas, this paper adopts the discrete transfer method(DTM) as the transmission method of the infrared radiation RTE.The discrete transfer method was first proposed by Lookwood to calculate the radiation transfer in the combustion chamber question.In this method, the boundary of microelements is used as the starting point of radiation transmission.By judging whether microelements and detection points are visible, if visible, a characteristic ray will be emitted to the detection point, and the infrared radiation emitted by each microelement will be transmitted along this characteristic ray, as shown in Figure 4. Finally all the characteristic rays received by the detection point are superimposed to obtain the total infrared radiation intensity at a certain detection point.When using the discrete transfer method to calculate the infrared radiation transfer problem, the microelement boundary is used as the starting point of radiation transfer, and the effective radiance of the microelement boundary is the focus of calculating the initial radiation.In the process of object A radiating outward, the external effective infrared radiation brightness of object A is composed of two parts, one part is the outward infrared radiation of object A itself, and the other part is the infrared radiation produced by other objects to the infrared radiation illuminance of object A .For the multistream serpentine nozzle, the flow channel profile is S-shaped, and the high-temperature components in front of the nozzle are blocked, but due to the curved configuration, a certain micro-element receives a large amount of infrared radiation illumination from other micro-elements.
For the multi-stream serpentine nozzle, assuming that there are N solid boundary microelements and M gas boundary microelements, then for microelement surface a, assuming that other microelements face a is visible, the effective radiation of microelement surface a The brightness Le(a) is： ( ) ( ) (2) The first item on the right side of the equation is the outward infrared radiance of micro-element surface A itself, and the second item on the right is the sum of infrared radiance produced by other micro-element faces B on micro-element A. The effective radiance is shown in Figure 5.In formula (2), a ε is the emissivity of the micro-element surface A, and a ρ is the reflectivity of the micro-element surface A. For it, on the one hand, it comes from the infrared radiation of the micro-element surface B itself, and on the other hand, it comes from the reflection of the infrared radiation illuminance of other micro-elements on the micro-element B. When the micro-element surface is a gas boundary microelement, the effective radiance of the wall is 0, so we have: ( ) ] The equations of eq k and eq γ δ    are bellow: When calculating the infrared radiation intensity, this paper divides the nozzle outlet into a horizontal detection surface and a vertical detection surface, as shown in Figure 6.In this paper, Lowtran7 is used to calculate the atmospheric transmittance characteristics.When calculating the infrared radiation of the multi-stream Serpentine nozzle, high-temperature components such as the low-pressure turbine upstream of the nozzle are considered, so the inlet of the nozzle is simplified as a gray body hot wall, and the emissivity is 0.9 , the temperature is given by the boundary conditions at the entrance, and the emissivity of other solid thermal walls in this study is set to ε=0.9.The infrared radiation on the outer wall of the nozzle is not considered when calculating the infrared radiation of the multi-stream Serpentine nozzle.Infrared radiation intensity at azimuth angles 0°, 5°, 10°, 15°, 20°, 30°, 40°, 60° and 90° on the detection surface.The IRSL characteristic value of IRSL in this paper is the average of infrared radiation intensity at various detection angles, and it is normalized by its max value.

Radial basis function
In the multi-objective optimization of the multi-stream Serpentine nozzle, it is necessary to first establish a mathematical model of the design parameters and performance parameters of the multi-stream Serpentine nozzle by selecting and calculating sample points.When selecting sample points, that is, experimental design, it is hoped to reduce the number of sample points as much as possible, so as to reduce the workload of accurate model analysis and calculation.At the same time, in order to obtain a reliable approximate model, the number of sample points must meet the needs of approximate modeling.At present, the commonly used experimental design methods include Latin hypercube sampling, uniform experimental design, and orthogonal experimental design.This paper adopts the orthogonal experimental design method to select sample points, which is a experimental design method summed up on the basis of a large number of practices, and has the characteristics of fast and high efficiency.Orthogonal experimental design can rationally arrange experiments by selecting an appropriate orthogonal design table according to the determined experimental factors (namely design variables) and the number of levels (that is, how many values the factors take).In this paper, the U21 (21 7 ) uniform design table is used to design the experimental scheme with 3 factors and 21 levels.According to the usage table of U21(21 7 ), select the first column, the third column and the fourth column in the U21(21 7 ) uniform design table to design the experimental scheme.The multi-stream Serpentine nozzle test samples and aerodynamic/infrared target response values obtained according to the orthogonal experimental design are shown in Table 2 After obtaining the test samples, it is necessary to establish an approximate model between the aerodynamic performance and infrared radiation characteristics of the multi-stream Serpentine nozzle and its design parameters, while the radial basis neural network does not require any mathematical model, and can be established according to the input and output data Approximate models, similar to black box theory, can be used to deal with nonlinear and ambiguous data, which can meet the requirements of conducting simulation experiments.In this paper, the Sigmoid function is used as the radial basis function to establish a three-layer feed-forward neural network composed of the input layer, the hidden layer and the output layer.The constant δ is used for optimization.The cross-validation technique is used to optimize the expansion constant δ .

Particle swarm optimization
The particle swarm optimization algorithm was proposed by Kennedy et al.It is a stochastic parallel optimization algorithm with simple structure and easy implementation.The basic concept of particle swarm optimization algorithm comes from the study of bird predation.When a flock of birds is looking for food, if one bird finds food, the rest of the birds will gather next to the bird that found the food.So each bird in the flock is regarded as a particle, and each particle is the potential feasible solution of "food", which is the optimal solution.The particle swarm optimization algorithm, similar to the genetic algorithm, is an iterative-based optimization technique.Particle swarm searches for the global optimal solution in N-dimensional space.In each iteration, each particle adjusts its search position and direction through the best position (pbest) and the best position in the entire population (gbest) experienced by the individual during the search process.Figure 8 shows a schematic diagram of particle velocity and position update when particles are iterated from generation n to generation n+1.Among them, xn and xn+1 represent the position of the particle at generation n and generation n+1 respectively, v1 represents the speed of the particle itself, v2 represents the speed at which the particle moves to the best position pbest experienced by itself, and v3 represents the particle's movement to the first generation The speed at which the best position gbest moves in the n generation population.Under the action of v1, v2 and v3, the particle finally moves from xn to xn+1.In the optimization process, it is assumed that when the particle swarm is iterated to the nth generation, the position of particle i is ( ) ( , ,..., ) , and the speed is ( ) ( , ,..., ) . Then at generation n+1, the velocity () i vn +1 and position () i xn +1 of the particle can be expressed as: x t x t v t + = + + 11 (7) Among them, the three items on the right side of the equal sign in formula 6 correspond to v1, v2 and v3 in Figure 8 respectively.w means weight.c1 and c2 represent the acceleration constants of the particles, usually in the range [0, 2].r1 and r2 represent random numbers uniformly distributed between [0, 1].
For multi-objective optimization problems, the core of multi-objective particle swarm optimization algorithm is to select the global optimal solution gbest.For multi-objective optimization problems, the number of objective functions is greater than 1, and the optimal solution of a certain objective function is not necessarily the optimal solution compared to other functions.There are certain conflicts.Therefore, there is a set of balanced solution sets for multi-objective optimization problems , and its optimal solution is called Pareto solution.For the multi-stream Serpentine nozzle aerodynamic/infrared multiobjective optimization problem studied in this paper, the purpose is to find the point with the best aerodynamic performance and the weakest infrared radiation intensity in the design space, but the design parameters with the best aerodynamic performance are Infrared radiation intensity is the worst, so a set of balanced solutions must be obtained.For the multi-objective optimization problem studied in this paper, its mathematical description is as follows:

Results and discussion
Before multi-objective optimization, this paper first analyzes the influence of various factors on aerodynamic/infrared.The experimental design using the uniform design method cannot be handled by the general analysis of variance, so this paper uses the method of multiple regression analysis to analyze the results to obtain the influence relationship between multiple factors and the target response value.Since the interaction between cubic items and factors with more than three factors is generally not considered in the analysis of the results of the uniform design method, the response function of multiple regression can be expressed as: In the formula: M -is the total number of factors; j x 2 -the quadratic term of factor j.
In Equation 9, reflects the interaction between factors j and k.When the interaction between factors is not considered, Equation 9 can be further simplified as: In the aerodynamic/infrared multi-objective optimization research of multi-stream Serpentine nozzle, without considering the interaction between various factors, the following formula can be used to describe the relationship between the target response and the factors: (12) Through the uniform design test data in Table 8, the analysis of variance results of the regression equations for different objective functions (Cfg and IRSL) can be easily obtained by using statistical data analysis software, as shown in Table 3.Among them, the F value is the result of the variance analysis, indicating the test of the entire regression equation, and Sig is the significance test value corresponding to the F value.It can be seen that for both Cfg and IRSL objective functions, Sig is less than 0.05.Therefore, it can be considered that the established regression equation is effective, and the regression equation can reflect the relationship between factors and responses to a certain extent.Since the absolute value of the regression coefficient bj is related to the unit of the corresponding factor, in order to compare the influence of each factor on the response value, the unit of each factor must be standardized to obtain a standardized regression coefficient j b .The larger the absolute value of j b , the greater the influence of the corresponding factor on the response.Table 4 shows the regression coefficients bj and standard regression coefficients j b of the regression equations for the two objective functions of Cfg and IRSL.It can be seen that for the objective function Cfg there is b b b 2 , therefore, the primary and secondary relationship of the three factors is 2 , that is, under the same standard, the change of nozzle drop pressure ratio has the greatest impact on the thrust coefficient, while the LBD has the weakest impact.For the objective function 2 , therefore, the primary and secondary relationship of the three factors is 2 , that is, under the same standard, the change of nozzle NPR has the greatest impact on the infrared radiation intensity, while the LBD has weakest impact.The aerodynamic/infrared multi-objective optimization process of the multi-stream Serpentine nozzle is mainly divided into two steps, approximate modeling and multi-objective optimization.In the process of establishing an approximate model for the aerodynamic performance target Cfg and the infrared target IRSL using radial basis neural network technology, the expansion coefficient δ is firstly optimized in the range of 0.01-2, and the step size is 0.01. Figure 9 shows the variation of the average error of the sample target response value with δ in the process of δ optimization.It can be seen that for the target Cfg, when δ=1.15, the average error of the samples is the smallest.For the target IRSL, when δ = 0.23, the average error of the samples is the smallest.Therefore, when establishing an approximate model, δ=1.15 and δ=0.23 are taken for the target Cfg and IRSL, respectively.After establishing the approximate model between the optimization objective and the optimal design parameters of the multi-stream Serpentine nozzle, the aerodynamic/infrared multi-objective optimization of the multi-stream Serpentine nozzle is carried out by using the particle swarm multiobjective optimization algorithm.During the optimization process, the particle swarm population size is 800, the external file size is 200, and the maximum number of iterations is 5000.Figure 10and Figure 11 respectively show the distribution of samples and optimized Pareto solutions in design space and target space.It can be seen from Fig10 that the three optimized parameters of the multi-stream Serpentine nozzle converge to one area.Table 9 shows the value range of the optimized Pareto solution design parameters.It can be seen from Fig11 that the Pareto solution moves toward the direction of increasing Cfg and decreasing IRSL, which meets the requirements of the optimization goal of high aerodynamic performance and low infrared radiation intensity for the multi-stream Serpentine nozzle.5.According to the design parameters of the Pareto solution to be verified, the modeling, grid division and numerical simulation calculation of the multi-stream S-bend nozzle are carried out, and the objective function values of the two verification points are obtained, as shown in Table 6. Figure 12 shows the positional relationship between the two verification points, the test samples and the Pareto solution set in the target space.It can be seen that due to the influence of the approximate model error, there is a slight deviation between the verification point and the Pareto solution set, but the two verification points are distributed near the Pareto solution set, and the coincidence is good.

Conclusion
Based on the radial basis neural network technology, this paper establishes an approximate model between the design parameters of the three-duct S-bend nozzle for adaptive engines and the aerodynamic design objectives and infrared design objectives, and performs regression for the influence of different parameters on the infrared radiation intensity According to the property analysis, the pressure ratio has the greatest influence on the infrared radiation intensity, the aspect ratio has a moderate influence, and the aspect ratio has the least influence on the length.After analyzing the influence of different design parameters on the infrared radiation intensity, a multi-objective particle swarm optimization algorithm was established, and the commonly used test functions were tested.The aerodynamic/infrared multiobjective optimization design with the infrared radiation intensity as the target, in which the S-shaped exhaust system for an adaptive engine with high aerodynamic performance and low infrared radiation intensity optimized has a pressure ratio between 4.01 and 4.8, and an aspect ratio between 4.01 and 4.8.Between 2.34 and 2.5, and the aspect ratio is between 6.06 and 6.9.When designing, a trade-off should be made between the optimization goals from the perspective of aerodynamics and infrared.

Figure 1 .
Figure 1.Structure of multi-stream Serpentine nozzle.At present, researchers' research on the multi-stream Serpentine nozzle mainly focuses on the flow and noise characteristics of the multi stream or Serpentine nozzle.Magstadt[11] of Syracuse University in the United States designed a multi-ply Multi-Aperture Rectangular Single Expansive Ramp Nozzle (MARS), and its aeroacoustic characteristics were studied through experiments.Spectrum Energy Company Rusher[12] studied the influence of mainstream heating on the flow structure and far field effect of noise through numerical simulation.Stack[13] of the Ohio State University at Austin used the LES of the turbulent inlet boundary to analyze the flow characteristics of the wake jet, shock wave and shear layer in detail under design conditions, and Hromisin[14] of the Pennsylvania State University investgated the aeroacoustic characteristics of the axisymmetric single-flow convergent-divergent nozzle and the multi-flow supersonic nozzle are studied under the same thrust with the experiment.For the multi-stream supersonic nozzle, the researchers studied its flow field characteristics, aeroacoustic characteristics and unsteady flow characteristics through numerical simulation and experiments[15][16][17][18][19][20][21] .As for serpentine nozzle ,researchers carried out a variety of design methods for the Serpentine nozzle[22][23][24] , and a lot of research on the flow characteristics of Serpentine nozzles, focusing on thrust vector performance and flow characteristics[25][26] , less research on the infrared radiation characteristics of multi-flow Serpentine nozzles.The structure of the multi-stream Serpentine nozzle is more complicated than that of the general Serpentine nozzle, and there are more configuration parameters.Different structural parameters will affect the aerodynamic performance and infrared radiation characteristics of the multi-stream Serpentine nozzle.In the design process of multi-stream Serpentine nozzle, the design goals of its aerodynamic performance and infrared stealth performance are often contradictory.In order to obtain the best aerodynamic performance, when designing the multi-stream Serpentine nozzle, it is necessary to minimize the change of the flow direction of the airflow in the nozzle, that is, to minimize the lengthto-diameter ratio of the multi-stream Serpentine nozzle and the Aspect ratio to reduce localized losses along the airflow.However, the change of length-to-diameter ratio and aspect ratio will affect the wall temperature distribution and gas mixing intensity of the nozzle, thus affecting the infrared radiation characteristics of the multi-stream Serpentine nozzle.Different design parameters will inevitably change the infrared radiation characteristics while affecting the aerodynamic performance.Aiming at this (a) The multi-stream serpentine nozzle.(b)The parameters of serpentine nozzle.

Figure 2 .
Figure 2. The structure and parameters of multi-stream serpentine nozzle.

Figure 4 .
Figure 4.The introduction of RTE.When using the discrete transfer method to calculate the infrared radiation transfer problem, the microelement boundary is used as the starting point of radiation transfer, and the effective radiance of the microelement boundary is the focus of calculating the initial radiation.In the process of object A radiating outward, the external effective infrared radiation brightness of object A is composed of two parts, one part is the outward infrared radiation of object A itself, and the other part is the infrared radiation produced by other objects to the infrared radiation illuminance of object A .For the multistream serpentine nozzle, the flow channel profile is S-shaped, and the high-temperature components in front of the nozzle are blocked, but due to the curved configuration, a certain micro-element receives a large amount of infrared radiation illumination from other micro-elements.For the multi-stream serpentine nozzle, assuming that there are N solid boundary microelements and M gas boundary microelements, then for microelement surface a, assuming that other microelements face a is visible, the effective radiation of microelement surface a The brightness Le(a) is： ( ) ( )

Figure 5 .
Figure 5.The effective radiance received from A. The relationship between the effective infrared radiation of microelement A and other visible microelements B can be obtained by combining Equation 2 and Equation 3, and the effective infrared radiance of each wall can be obtained by solving the large linear matrix equation after the simultaneous connection。Considering the inhomogeneity of gas composition parameters in each micro-element unit and its connected unit, this paper uses the Malkmus narrow-band model (SNB) combined with Curtis-Godson approximation to calculate the spectral transmittance.The bandwidth of the narrow-band model is 25cm -1 , the formula for calculating the transmittance under a certain bandwidth by the narrow band model is shown in formula (3), where γ is the average pressure absorption coefficient, k is the average half-width of spectral lines, and β is the average spacing of spectral lines.

Figure 6 .
Figure 6.The Schematic diagram of Probe points and plane.
the best δ in the range 0.01~2 and the step size is 0.01.Finally the optimal expansion constant δ=0.15 is obtained.The effect of the approximate model is shown in the figure7.The results obtained by the approximate model are in good agreement with the theoretical value, and basically coincide with the actual curve.

Figure 8 .
Figure 8. Schematic diagram of the iterative process of particles.In the optimization process, it is assumed that when the particle swarm is iterated to the nth generation, them, represents target response Cfg or IRSL, and represent factors NPR, LBD and AR respectively.11 can be linearized as: y b b x b x b x b x b x b x

Figure 9 .
Figure 9. Optimization process for extended constant δ.After establishing the approximate model between the optimization objective and the optimal design parameters of the multi-stream Serpentine nozzle, the aerodynamic/infrared multi-objective optimization of the multi-stream Serpentine nozzle is carried out by using the particle swarm multiobjective optimization algorithm.During the optimization process, the particle swarm population size is 800, the external file size is 200, and the maximum number of iterations is 5000.Figure10andFigure11respectively show the distribution of samples and optimized Pareto solutions in design space and target space.It can be seen from Fig10 that the three optimized parameters of the multi-stream Serpentine nozzle converge to one area.Table9shows the value range of the optimized Pareto solution design parameters.It can be seen from Fig11 that the Pareto solution moves toward the direction of increasing Cfg and decreasing IRSL, which meets the requirements of the optimization goal of high aerodynamic performance and low infrared radiation intensity for the multi-stream Serpentine nozzle.

Figure 10 .
Figure 10.Distribution of particles over design space.

Figure 11 .
Figure 11.Distribution of particles in target space.
In order to verify the reliability of the optimization results, two Pareto solutions are selected in the Pareto solution set for verification.The design parameters of the two selected Pareto solutions are shown in Table

Table 1 .
The range of design parameters.

Table 2 .
: Multi-stream Serpentine nozzle test samples and aerodynamic/infrared target response values

Table 3 .
Analysis of variance table for regression equations.

Table 4 .
Regression coefficients and standard regression coefficients for regression equations.

Table 5 .
Range of design parameters for Pareto solutions.

Table 6 .
Verifying the design parameters of the Pareto solution.

Table 7 .
Objective function values for verification points.

Table 8 .
Verify the point numerical simulation function values.